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Group Definitions and Examples Homework

Directions: Write your homework legibly with a pencil. Complete the homework on time and by yourself. For each problem, write the instructions, label the solution/proof and show all steps. Do not turn in your scratch work. Staple your pages together, with the problems in the correct order, and use this page as a cover sheet with your name at the top right.

(1) Show that is a group with respect to multiplication.

(2) Show that is a group with respect to addition.

(3) Show that is a group with respect to multiplication.

(4) Show that the set of all mappings from to is a group with respect to function addition.

(5) Show that is a group with respect to function multiplication.

(6) Prove that if is a group, and for some then must be the identity element of

(7) Let be the rational numbers whose denominators are a power of the prime when reduced to lowest terms. Prove that is a group under

(8) Let be a group with operation and let Prove that is a group under

(9) Prove that if for all in the group then is Abelian.

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